Modelo esférico quântico elementar

Abstract

As propriedades termodinâmicas do modelo esférico médio do ferromagnetismo, na versão de Curie-Weiss, que inclui interações entre todos os pares de variáveis de spin, podem ser obtidas de maneira exata e analisadas de forma particularmente simples e pedagógica. Torna-se então interessante considerar uma versão quântica desse modelo, que vamos denominar "modelo esférico quântico elementar ", e que também pode ser analisada detalhadamente, em termos da temperatura T e de um parâmetro g associado às flutuações quânticas. Esse sistema proporciona um dos exemplos mais simples de uma transição de fase quântica. Mantendo o estilo pedagógico, fazemos contato com diversos resultados da literatura e apresentamos comentários sobre certas questões, como a correção de anomalias do comportamento clássico e o papel do limite termodinâmico no estabelecimento de uma singularidade do "tipo Bose-Einstein".

Quantum phase transition in the spin-anisotropic quantum spherical model

“…This makes it clear that interactions between the momenta cannot always be absorbed into a change of variables. 10 In figure 3, we compare the shape of the critical line g c = g c (λ), normalised to the value at g c (0) at λ = 0, of the bosonic quantum spherical model (1.11), with the fermionic quantum XY model. In 1D, the latter model reduces to free fermions.…”

“…The properties of this exactly solvable model have been analysed in great detail, see e.g. [57,64,56,68,18,62,11,32,58,9,10]. Independently, the quantum spherical model was also obtained via the so-called 'hamiltonian limit' [50] as the logarithm of the transfer matrix in an extremely anisotropic limit [65,35,38].…”

Axiomatic construction of quantum Langevin equations

“…These are equivalent to the averages given in (1.6). 8 Hence, the qfdt implies non-markovian noise correlators.…”

“…distinct from mean-field theory) of this model follows from the coupling of these modes by the so-called spherical constraint, which is included into the qle via a time-dependent Lagrange multiplier. The equilibrium critical behaviour of the quantum spherical model is well-known [51,40,50,66,52,58,7,8,67]: while for finite temperatures T > 0, the critical behaviour is the same as for the classical spherical model [6,44], at temperature T = 0 there is a quantum critical point in the same universality class as the (d + 1)-dimensional classical spherical model [42]. The time-dependent non-equilibrium classical system has also been very thoroughly analysed, see e.g.…”

“…Since the ν → ∞-behaviour in (3.25) is less singular than in (3.28a), it does not depend on an eventual cut-off, see[31, sec. 3.3] 8. In appendix B, we shall study in more detail the essential singularities of the average anti-commutator {η s (t), η p (t ′ )} = λ 2π J ν 2T , t − t ′ , with J(a, τ ) := R dν coth (aν) sin (ντ ).…”